System for analysing the frequency of resonating devices

ABSTRACT

The general field of the invention is that of resonating or vibrating devices. One of the tricky points with this type of device is that of correctly measuring the vibration frequency in a disturbed environment. The analysis system according to the invention is based on the use of the Hilbert transform of a function U representative of the position A of the vibrations of the device. More precisely, in its basic version, the analysis system comprises means making it possible to realize the second derivative of the function U denoted U (2) , the Hilbert transform of the function U denoted V, the second derivative of this transform denoted V (2)  as well as a function equal to 
     
       
         
           
             
               - 
               
                 
                   
                     U 
                     · 
                     
                       U 
                       
                         ( 
                         2 
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                   + 
                   
                     V 
                     · 
                     
                       V 
                       
                         ( 
                         2 
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                     U 
                     2 
                   
                   + 
                   
                     V 
                     2 
                   
                 
               
             
             , 
           
         
       
     
     this latter function being mathematically equal to the function 
     
       
         
           
             
               ω 
               1 
               2 
             
             - 
             
               
                 ρ 
                 1 
                 
                   ( 
                   2 
                   ) 
                 
               
               
                 ρ 
                 1 
               
             
           
         
       
     
     with ω 1  the instantaneous angular frequency of U, ρ 1  the instantaneous amplitude of U, ρ 1   (2)  the second derivative of said instantaneous amplitude and being representative of the square of the resonant angular frequency Ω of the vibrating device. Several variants of this initial device are proposed.

RELATED APPLICATIONS

The present application is based on, and claims priority from, FrenchApplication Number 07 03733, filed May 25, 2007, the disclosure of whichis hereby incorporated by reference herein in its entirety.

FIELD OF THE INVENTION

The field of the invention is that of resonating or vibrating devices.It is known that the principle of measurement of a large number ofsensors is based on measuring the frequency of the oscillations of amechanical system oscillating either under free oscillations, or underforced oscillations, this frequency depending on the parameter that oneseeks to measure. Mention may be made, by way of example, of vibratingbarrel accelerometers also called VBAs used to measure accelerations, itbeing understood that what follows may be readily generalized to anyvibrating system.

DESCRIPTION OF THE PRIOR ART

FIG. 1 depicts a basic diagram of an accelerometer of this type. Itessentially comprises a mass connected to two identical and parallelbeams 1 forming a tuning fork, each beam carries two electrodes 2 and 3,the first electrode 2 serving for excitation and the second electrode 3for detection. Under the effect of an acceleration along the sensitiveaxis x of the accelerometer, the mass moves in translation along thissensitive axis thereby causing an extension or contraction of the tuningfork and thus modifying its resonant frequency according to relation 0below.

$\begin{matrix}{f_{a} = {f_{0}\sqrt{1 + \frac{a}{\gamma_{C}}}}} & {{Relation}\mspace{14mu} 0}\end{matrix}$

with f_(a): resonant frequency of the accelerometer subject to theacceleration,

f₀: resonant frequency of the sensor at rest, that is to say subject toa zero input acceleration,

a: acceleration applied to the sensor,

γ_(c): critical buckling acceleration, a physical parameter of theresonator.

The principle of measuring the frequency is as follows. The excitationand vibration of the beams are sustained by the excitation electrode 2covering a part of the length of each beam. The excitation voltage isthe sum of a continuous voltage V₀ and of an alternating voltage v thatalternates at the resonant frequency of the oscillator, created byfeedback of the sensor output signal. The remainder of the surface ofthe two beams is covered by a detection electrode 3 polarized to thevoltage V₀. The vibration motion of the beams causing the distancebetween the beams and this electrode to vary, a detection currentappears in the electrode. This current passes into a charge amplifyingfilter 5. The voltage at the output of this filter constitutes thesensor output signal. This output signal is used on the one hand for theprocessing 6 making it possible to retrieve, from the resonant frequencymeasurement, the acceleration applied to the sensor and on the otherhand as input into the feedback loop 4 making it possible to generatethe alternating excitation voltage.

On the basis of the signals returned by the accelerometer, it isnecessary to retrieve the acceleration applied to the sensor. Thenonlinear nature of relation 0 poses a fundamental problem. Indeed, theacceleration a in relation 0 contains not only the useful accelerationapplied to the sensor but also vibration terms that may reach very highvalues. The mass-spring system formed by the mass and the resonatorexhibits a resonant frequency f_(R). At this frequency, noise is greatlyamplified. The acceleration applied to the sensor comprises alow-frequency term whose band typically lies over an interval varyingfrom 0 Hz to 400 Hz which is the static or dynamic acceleration that onewishes to determine by virtue of the processing and a term centred onthe frequency f_(R) which typically lies between 3 kHz and 5 kHz whichcorresponds to the noise filtered by the mass-spring system and whichforms parasitic vibrations that may degrade the results of theprocessing. These terms give rise both to a bias error and a scalefactor error at the level of the acceleration. The level of thevibrations being random, these errors cannot be compensated for. Ofcourse, these problems are common to any device subject to a parasiticvariation of its input quantity.

The motion of the beams may be represented by an equivalentmonodimensional oscillator along the direction and the sense of thesensitive axis of the sensor. The position denoted A of this oscillatorsatisfies the following differential equation denoted relation 1:

$\begin{matrix}{{{\frac{^{2}A}{t^{2}} + {\frac{\omega_{0}}{Q_{R}}\frac{A}{t}} + {{\omega_{0}^{2}\left( {1 + \frac{a}{\gamma_{C}}} \right)}A} + {\beta_{i}A^{3}}} = {e(t)}}} & {{Relation}\mspace{14mu} 1}\end{matrix}$

with ω₀ the rest resonant angular frequency of the sensor, under zeroacceleration,

Q_(R): quality factor of the resonator,

γ_(c): critical buckling acceleration of the sensor,

a: acceleration applied to the sensor. This acceleration comprises boththe low-frequency term that one seeks to measure and high-frequencyterms due to the vibrations,

β_(i): nonlinearity coefficient of order 3,

e(t): time dependent excitation term.

$\omega_{0}\sqrt{1 + \frac{a}{\gamma_{C}}}$

is the resonant angular frequency of the vibrating system subjected tothe acceleration a.

Generally, use is made of two resonators having sensitive axes of likedirection and opposite sense making it possible to perform themeasurement on two different pathways. The following two equations thenhold: using the same notation as previously, the index 1 referring topathway 1 and the index 2 to pathway 2.

Relation  2  on  pathway  1${\frac{^{2}A_{1}}{t^{2}} + {\frac{\omega_{0,1}}{Q_{R\; 1}}\frac{A_{1}}{t}} + {{\omega_{0,1}^{2}\left( {1 + \frac{a}{\gamma_{C,1}}} \right)}A_{1}} + {\beta_{i,1}A_{1}^{3}}} = {e_{1}(t)}$Relation  3  on  pathway  2${\frac{^{2}A_{2}}{t^{2}} + {\frac{\omega_{0,2}}{Q_{R\; 2}}\frac{A_{2}}{t}} + {{\omega_{0,2}^{2}\left( {1 - \frac{a}{\gamma_{C,2}}} \right)}A_{2}} + {\beta_{i,2}A_{2}^{3}}} = {e_{2}(t)}$

The current in the detection electrode is proportional to the speed ofthe equivalent oscillator. The transfer function of the chargeamplifying filter is that of an integrator. The voltage output by thecharge amplifier is obtained by integrating the speed of the equivalentoscillator. Though the signal corresponding to the position of theoscillator may comprise a low-frequency term, this term is not foundagain at the output of the charge amplifier. The voltage output by thecharge amplifier is then proportional to the high-frequency terms of theoscillator position signal.

Denoting by U₁ and U₂ the voltages output by the filter on pathways 1and 2, the following relations hold:

U₁=K₁I₁ and U₂=K₂I₂ with K₁ and K₂ known constants, depending on themechanical and electrical parameters of the resonators and the chargeamplifier, I₁ and I₂ representing the high-frequency parts of A₁ and A₂.

To determine the acceleration from the frequency, a first procedureconsists in considering that the frequency denoted f of the sensoroutput signal is connected to the acceleration by the relation

$f \approx {f_{0}{\sqrt{1 + \frac{a}{\gamma_{C}}}.}}$

Denoting by f₁ and f₂ the frequencies of the output signals on pathways1 and 2 and by carrying out a finite expansion of said frequencies, thefollowing relations are obtained:

$\begin{matrix}{f_{1} \approx {f_{0,1}\sqrt{1 + \frac{a}{\gamma_{C,1}}}} \approx {f_{0,1}\begin{pmatrix}{1 + \frac{a}{2\gamma_{C,1}} -} \\{\frac{a^{2}}{8\gamma_{C,1}^{2}} + \frac{a^{3}}{16\gamma_{C,1}^{3}} + \ldots}\end{pmatrix}}} & {{Relation}\mspace{14mu} 4} \\{f_{2} \approx {f_{0,2}\sqrt{1 - \frac{a}{\gamma_{C,2}}}} \approx {f_{0,2}\begin{pmatrix}{1 - \frac{a}{2\gamma_{C,2}} -} \\{{\frac{a^{2}}{8\gamma_{C,2}^{2}} - \frac{a^{3}}{16\gamma_{C,2}^{3}} + \ldots}\;}\end{pmatrix}}} & {{Relation}\mspace{14mu} 5}\end{matrix}$

The acceleration is decomposed:

-   -   into a first low-frequency useful part;    -   into a second part due to the vibrations at the resonant        frequency of the mass-spring mode. The following are obtained        respectively on pathways 1 and 2:

a=a ₀+α_(V,1) cos(2πf _(R) t)

a=a ₀+α_(V,2) cos(2πf _(R) t)

The acceleration may be estimated by virtue of a differentialprocessing. Only the low-frequency terms of the difference of thefrequencies are of interest. This yields the following relation:

$\frac{f_{1} - {\frac{f_{0,1}}{f_{0,2}}f_{2}}}{f_{0,1}} = {{a_{0}\left( {\frac{1}{2\gamma_{C,1}} + \frac{1}{2\gamma_{C,2}}} \right)} + \frac{\alpha_{0}^{2} + \frac{\alpha_{V,2}^{2}}{2}}{8\; \gamma_{C,2}^{2}} - \frac{\alpha_{0}^{2} + \frac{\alpha_{V,1}^{2}}{2}}{8\; \gamma_{C,1}^{2}} + \frac{\alpha_{0}^{3} + {3\; a_{0}\frac{\alpha_{V,1}^{2}}{2}}}{16\; \gamma_{C,1}^{3}} + \frac{\alpha_{0}^{3} + {3a_{0}\frac{\alpha_{V,2}^{2}}{2}}}{16\; \gamma_{C,2}^{3}}}$

This relation shows that:

-   -   the term of order 3 of the finite expansion not being eliminated        by the differential processing, it causes a nonlinearity error        corresponding to the term in a₀ ³ and a scale factor error        corresponding to the term in a₀α_(V) ². The amplitude of the        vibrations being unknown, this scale factor error cannot be        compensated.    -   The term of order 2 is only partly reduced by the differential        processing. There is therefore also a nonlinearity error        corresponding to the term in a₀ ² and a bias error corresponding        to the term in α_(V) ². Here again, this bias error cannot be        compensated.

Another procedure consists in determining the acceleration from thedifference of the square of the frequencies. This procedure is describedin the French patent with the reference FR 2 590 991. The frequency ofthe sensor output signal is assumed to be connected to the accelerationby the following relations, with the same notation as previously:

${{For}\mspace{14mu} {pathway}\mspace{14mu} 1\mspace{14mu} f_{I,1}} \approx {f_{0,1}\sqrt{1 + \frac{a_{0} + {\alpha_{V,1}{\cos \left( {2\; \pi \; f_{R}t} \right)}}}{\gamma_{C,1}}}}$${{and}\mspace{14mu} {for}\mspace{14mu} {pathway}\mspace{14mu} 2\mspace{14mu} f_{I,2}} \approx {f_{0,2}\sqrt{1 - \frac{a_{0} + {\alpha_{V,2}{\cos \left( {2\; \pi \; f_{R}t} \right)}}}{\gamma_{C,2}}}}$

The device which makes it possible to extract the frequencies ismodelled in the form of a transfer function having a unit gain and azero phase shift for low frequencies, a gain β and a phase shift φ forthe frequency f_(R) and a zero gain for the frequencies that aremultiples of f_(R). This device may, for example, be a PLL, the acronymstanding for Phase-Locked Loop.

A finite expansion of order 3 then gives:

$\frac{f_{i,1}}{f_{0,1}} = {1 + \frac{\begin{matrix}{a_{0} + \alpha_{V,1}} \\{\cos \left( {2\; \pi \; f_{R}t} \right)}\end{matrix}}{2\gamma_{C,1}} - \frac{\begin{matrix}{a_{0}^{2} + {\alpha_{V,1}^{2}\cos^{2}\left( {2\; \pi \; f_{R}t} \right)} +} \\{2a_{0}\alpha_{V,1}{\cos \left( {2\pi \; f_{R}t} \right)}}\end{matrix}\;}{8\gamma_{C,1}^{2}} + {\frac{1}{16\; \gamma_{C,1}^{3}}\begin{bmatrix}{a_{0}^{3} + {\alpha_{V,1}^{3}{\cos^{3}\left( {2\; \pi \; f_{R}t} \right)}} + {3a_{0}\alpha_{V,1}^{2}}} \\{{\cos^{2}\left( {2\; \pi \; f_{R}t} \right)} + {3a_{0}^{2}\alpha_{V,1}{\cos \left( {2\; \pi \; f_{R}t} \right)}}}\end{bmatrix}}}$ and$\frac{f_{i,2}}{f_{0,2}} = {1 - \frac{\begin{matrix}{a_{0} + \alpha_{V,2}} \\{\cos \left( {2\; \pi \; f_{R}t} \right)}\end{matrix}}{2\gamma_{C,2}} - \frac{\begin{matrix}{a_{0}^{2} + {\alpha_{V,2}^{2}\cos^{2}\left( {2\; \pi \; f_{R}t} \right)} +} \\{2a_{0}\alpha_{V,2}{\cos \left( {2\pi \; f_{R}t} \right)}}\end{matrix}\;}{8\gamma_{C,2}^{2}} - {\frac{1}{16\; \gamma_{C,2}^{3}}\begin{bmatrix}{a_{0}^{3} + {\alpha_{V,2}^{3}{\cos^{3}\left( {2\; \pi \; f_{R}t} \right)}} + {3a_{0}\alpha_{V,2}^{2}}} \\{{\cos^{2}\left( {2\; \pi \; f_{R}t} \right)} + {3a_{0}^{2}\alpha_{V,2}{\cos \left( {2\; \pi \; f_{R}t} \right)}}}\end{bmatrix}}}$

At the output of the frequency estimation device, we have, in accordancewith the above assumptions:

$\frac{f_{s,1}}{f_{0,1}} = {1 + \frac{\begin{matrix}{a_{0} + {\beta\alpha}_{V,1}} \\{\cos \left( {{2\; \pi \; f_{R}t} + \varphi} \right)}\end{matrix}}{2\gamma_{C,1}} - \frac{\begin{matrix}{a_{0}^{2} + {\alpha_{V,1}^{2}/2} + {2a_{0}\alpha_{V,1}}} \\{{{\beta cos}\left( {{2\; \pi \; f_{R}t} + \varphi} \right)}\;}\end{matrix}}{8\gamma_{C,1}^{2}} + {\frac{1}{16\; \gamma_{C,1}^{3}}\begin{bmatrix}{a_{0}^{3} + {\alpha_{V,1}^{3}\frac{3}{4}{{\beta cos}\left( {{2\; \pi \; f_{R}t} + \varphi} \right)}} +} \\{{3a_{0}\frac{\alpha_{V,1}^{2}}{2}} + {3a_{0}^{2}\alpha_{V,1}{{\beta cos}\left( {{2\; \pi \; f_{R}t} + \varphi} \right)}}}\end{bmatrix}}}$ and$\frac{f_{s,2}}{f_{0,2}} = {1 - \frac{\begin{matrix}{a_{0} + {\beta\alpha}_{V,2}} \\{\cos \left( {{2\; \pi \; f_{R}t} + \varphi} \right)}\end{matrix}}{2\gamma_{C,2}} - \frac{\begin{matrix}{a_{0}^{2} + {\alpha_{V,2}^{2}/2} + {2a_{0}\alpha_{V,2}}} \\{{{\beta cos}\left( {{2\; \pi \; f_{R}t} + \varphi} \right)}\;}\end{matrix}}{8\gamma_{C,2}^{2}} - {\frac{1}{16\; \gamma_{C,2}^{3}}\begin{bmatrix}{a_{0}^{3} + {\alpha_{V,2}^{3}\frac{3}{4}{{\beta cos}\left( {{2\; \pi \; f_{R}t} + \varphi} \right)}} +} \\{{3a_{0}\frac{\alpha_{V,2}^{2}}{2}} + {3a_{0}^{2}\alpha_{V,2}{{\beta cos}\left( {{2\; \pi \; f_{R}t} + \varphi} \right)}}}\end{bmatrix}}}$

These two frequencies are squared and the low-frequency terms areretained.

$f_{S,1}^{2} = {f_{0,1}^{2}\left\lbrack {1 + \frac{a_{0}}{\gamma_{C,1}} + {\frac{\alpha_{V,1}^{2}}{8\; \gamma_{C,1}^{2}}\left( {\beta^{2} - 1} \right)} + {\frac{a_{0}\alpha_{V,1}^{2}}{8\; \gamma_{C,1}^{3}}\left( {1 - \beta^{2}} \right)}} \right\rbrack}$and$f_{S,2}^{2} = {f_{0,2}^{2}\left\lbrack {1 - \frac{a_{0}}{\gamma_{C,2}} + {\frac{\alpha_{V,2}^{2}}{8\; \gamma_{C,2}^{2}}\left( {\beta^{2} - 1} \right)} - {\frac{a_{0}\alpha_{V,2}^{2}}{8\; \gamma_{C,2}^{3}}\left( {1 - \beta^{2}} \right)}} \right\rbrack}$

By performing the differential processing, we obtain:

$\frac{f_{S,1}^{2} - {\frac{f_{0,1}^{2}}{f_{0,2}^{2}}f_{S,2}^{2}}}{f_{0,1}^{2}} = {{a_{0}\begin{pmatrix}{\frac{1}{\gamma_{C,1}} +} \\\frac{1}{\gamma_{C,2}}\end{pmatrix}} + {\frac{\left( {\beta^{2} - 1} \right)}{8}\begin{pmatrix}{\frac{\alpha_{V,1}^{2}}{\gamma_{C,1}^{2}} -} \\\frac{\alpha_{V,2}^{2}}{\gamma_{C,2}^{2}}\end{pmatrix}} + {\frac{a_{0}\left( {1 - \beta^{2}} \right)}{8}\begin{pmatrix}{\frac{\alpha_{V,1}^{2}}{\gamma_{C,1}^{2}} +} \\\frac{\alpha_{V,2}^{2}}{\gamma_{C,2}^{3}}\end{pmatrix}}}$

If the sensor is subject to vibrations and if the device for calculatingthe frequencies does not ensure wideband demodulation, that is to say ifβ is different from 1, a scale factor error and a bias error whichcannot be completely eliminated by the differential processing are againfound. These errors are not compensatable.

SUMMARY OF THE INVENTION

The system according to the invention does not exhibit the abovedrawbacks. It is known that the spectrum of the output signal is centredon the frequency f_(a0) equal to the sum of the central frequency of theresonator and of a deviation due to the low-frequency acceleration a₀applied to the sensor. This spectrum comprises lines situated at thefrequencies f_(a0)±k.f_(R) due to the vibrations. The system accordingto the invention implements an algorithm realizing a widebanddemodulation of the sensor output signal, taking into account the linesof significant amplitude. The algorithm is based on rigorous theoreticalrelations between the stiffness variation induced by the accelerationand the parameters characteristic of the sensor output signal which arethe instantaneous amplitude and the instantaneous frequency.Theoretically, the spectrum of the output signal contains lines atfrequencies below f_(R). In practice, given the frequencies and themodulation indices involved, these lines are of negligible amplitudes.It is therefore possible to consider that the spectral support of thesignal is disjoint from the vibration terms which are of lower frequencythan it. On account of these lines, the spectrum of the signal extendsover a wideband. It is necessary for the sampling frequency to be highenough to contain the spectrum of the signal. For example, with themodulation indices and the frequencies involved for the accelerometersat the level of the rest resonant frequency and vibrations, the order ofmagnitude of the sampling frequency must be 250 kHz.

More precisely, the subject of the invention is a system for analysingthe oscillation frequency of a device vibrating along an axis, saidsystem comprising means for measuring the position A of the device alongthis axis, the signal emanating from said means and representative ofthe position A being represented by a time dependent function U,characterized in that said system comprises first means making itpossible to calculate the function

${\omega_{1}^{2} - \frac{\rho_{1}^{(2)}}{\rho_{1}}},$

ω₁ being the instantaneous angular frequency of U, ρ₁ its instantaneousamplitude and ρ₁ ⁽²⁾ the second derivative of said instantaneousamplitude, the function

$\omega_{1}^{2} - \frac{\rho_{1}^{(2)}}{\rho_{1}}$

being representative of the square of the resonant angular frequency Ωof the vibrating device.

Advantageously, said first means comprise means for calculating thesecond derivative of the function U denoted U⁽²⁾, the Hilbert transformof the function U denoted V, the second derivative of this Hilberttransform denoted V⁽²⁾ as well as a first function equal to

$- \frac{{U \cdot U^{(2)}} + {V \cdot V^{(2)}}}{U^{2} + V^{2}}$

mathematically equal to

${\omega_{1}^{2} - \frac{\rho_{1}^{(2)}}{\rho_{1}}},$

representative of the square of the resonant angular frequency Ω of thevibrating device.

Advantageously, when the differential equation representative of thevariations in the position A of the device comprises nonlinearity termsof order 3, said device then comprises second means making it possibleto realize, using the previous notation, a second function equal to

${{- \frac{{V^{(2)}V} + {U^{(2)}U}}{U^{2} + V^{2}}} - {\beta_{i}\left\lbrack {\frac{3}{4\; K_{1}^{2}}\left( {U^{2} + V^{2}} \right)} \right\rbrack}},$

representative of the square of the resonant angular frequency Ω of thevibrating device, β_(i) and K₁ being constants.

Advantageously, when the vibrating device has forced oscillations, thatis to say the device comprises a phase- and amplitude-slaving disposedin such a way as to eliminate the natural damping of the device, saidslaving generating a polarization V₀, said device then comprises thirdmeans making it possible to realize, using the previous notation, athird function equal to

${{- \frac{{V^{(2)}V} + {U^{(2)}U}}{U^{2} + V^{2}}} - {\beta_{i}\left\lbrack {{\frac{3}{4\; K_{1}^{2}}\left( {U^{2} + V^{2}} \right)} + {3\left( \frac{K_{3}V_{0}^{2}}{- \frac{{UU}^{(2)} + {VV}^{(2)}}{U^{2} + V^{2}}} \right)^{2}}} \right\rbrack}},$

representative of the square of the resonant angular frequency Ω of thevibrating device, β_(i), K₁ and K₃ being constants.

Advantageously, the square of the resonant angular frequency is linkedto a parameter a to be measured by the relation:

$\Omega^{2} = {{\omega_{0}^{2}\left( {1 + \frac{a}{\gamma_{C}}} \right)}\gamma_{C}}$

being a constant, ω₀ being the initial angular frequency in the absenceof said parameter.

Advantageously, the vibrating device comprises two identical means ofvibration, each of the means being connected to the analysis system,said system comprising:

-   -   measurement means able to calculate on the one hand, either a        first function or a second function or a third function        representative of the square of the resonant angular frequency        Ω₁ of the first means of vibration and on the other hand either        a first function or a second function or a third function        representative of the square of the resonant angular frequency        Ω₂ of the second means of vibration;    -   means for calculating the following functions:

$\Omega_{1}^{2} - {\frac{\omega_{01}^{2}}{\omega_{02}^{2}}\Omega_{2}^{2}}$and$\Omega_{1}^{2} + {\frac{\omega_{0,1}^{2}}{\omega_{0,2}^{2}}\frac{\gamma_{C,2}}{\gamma_{C,1}}\Omega_{2}^{2}}$

using the same notation as previously, the indices representing thefirst or the second means of vibration.

Advantageously, the analysis system is an electronic system, thefunction U being an electrical parameter, said system comprising meansof digitizing and sampling the function U, first finite impulse responsefilters able to realize the Hilbert transform of an electronic function,second finite impulse response filters able to realize the derivative ofan electrical function, delay lines making it possible to synchronizethe various sampled signals, electronic means realizing the functions ofsummations, multiplication and division and band-stop filters andlow-pass filters.

Still other objects and advantages of the present invention will becomereadily apparent to those skilled in the art from the following detaileddescription, wherein the preferred embodiments of the invention areshown and described, simply by way of illustration of the best modecontemplated of carrying out the invention. As will be realized, theinvention is capable of other and different embodiments, and its severaldetails are capable of modifications in various obvious aspects, allwithout departing from the invention. Accordingly, the drawings anddescription thereof are to be regarded as illustrative in nature, andnot as restrictive.

BRIEF DESCRIPTION OF THE DRAWINGS

The present invention is illustrated by way of example, and not bylimitation, in the figures of the accompanying drawings, whereinelements having the same reference numeral designations represent likeelements throughout and wherein:

FIG. 1 represents the basic diagram of a vibrating-beams oscillator;

FIG. 2 represents the overview of the calculation according to theinvention of the square of the angular frequencies of the system;

FIG. 3 represents the overview of the calculation of the parameters tobe measured by differential processing.

MORE DETAILED DESCRIPTION

As stated, the core of the invention relies on an algorithm realizing awideband demodulation of the sensor output signal. This algorithmimplements the Hilbert transforms and their properties.

Let x(t) be a real signal, the Hilbert transform {circumflex over(x)}(t) of x(t) is defined by:

${\hat{x}(t)} = {{\frac{1}{\pi}{\int{\frac{x(u)}{t - u}{u}}}} = {\frac{1}{\pi}{\lim\limits_{H->0}\left( {{\int_{- \infty}^{t - H}{\frac{x(u)}{t - u}\ {u}}} + {\int_{t + H}^{\infty}{\frac{x(u)}{t - u}{u}}}} \right)}}}$

From the frequency point of view, if X(f) and {circumflex over (X)}(f)denote the respective Fourier transforms of {circumflex over (x)}(t) andx(t), we have:

{circumflex over (X)}(f)=−j sgn(f)X(f)

-   -   with sgn(f)=+1 for f>0 sgn(f)=−1 for f<0 and sgn(f)=0 for f=0

The analytic signal z_(x)(t) is defined by z_(x)(t)=x(t)+j{circumflexover (x)}(t). Its Fourier transform therefore equals, in view of theforegoing:

Z _(x)(f)=2X(f) if f>0,

Z _(x)(f)=0 if f<0

The spectrum not being symmetric, the signal z_(x)(t) is complex. We putz_(x)(t)=ρ(t)e^(jφ(t)). ρ(t) is the instantaneous amplitude of thesignal x, φ(t) is the instantaneous phase of the signal x

We then have

x(t)=ρ(t)cos(φ(t)) and {circumflex over (x)}(t)=ρ(t)sin(φ(t))

We also have ρ²=x²+{circumflex over (x)}²We define

${{\omega (t)} = \frac{\phi}{t}},$

the instantaneous angular frequency of the signal x

The following three properties of the Hilbert transform are usedsubsequently in the description:

-   -   When we have two signals x and y of disjoint spectral support,        with x of lower frequency than y, the Hilbert transform of the        product xy satisfies: H(xy)=xH(y).    -   Let x(t) be a signal of instantaneous amplitude ρ and of        instantaneous phase φ. It is assumed that ρ is of lower        frequency than cos(φ). The Hilbert transform of x² is then

$\frac{1}{2}{\rho^{2}(t)}{\sin \left( {2\; {\phi (t)}} \right)}$

and that of x³ is

$\frac{1}{4}{\rho^{3}\left\lbrack {{3\; {\sin (\phi)}} + {\sin \left( {3\; \phi} \right)}} \right\rbrack}$

-   -   Let I be a signal and Q its Hilbert transform, ω being the        instantaneous angular frequency of I, ρ being the instantaneous        amplitude of I. We then have, using I=ρ cos(φ) and Q=ρ sin(φ),        the following equality

${- \frac{{II}^{(2)} + {QQ}^{(2)}}{I^{2} + Q^{2}}} = {\omega^{2} - \frac{\rho^{(2)}}{\rho}}$

The position denoted A₁ of a first oscillator satisfies the followinggeneral differential equation, with the same notation as previously:

${{{Relation}\mspace{14mu} 2\text{:}\mspace{11mu} \frac{^{2}A_{1}}{t^{2}}} + {\frac{\omega_{0,1}}{Q_{R,1}}\frac{A_{1}}{t}} + {{\omega_{0,1}^{2}\left( {1 + \frac{a}{\gamma_{C,1}}} \right)}A_{1}} + {\beta_{i,1}A_{1}^{3}}} = {e_{1}(t)}$

In the case where this resonator does not comprise any third-ordernonlinearity term and in the case where the excitation term merelycompensates exactly for the damping term, then relation 2 simplifies andbecomes relation 2bis:

${\frac{^{2}A_{1}}{t^{2}} + {{\omega_{0,1}^{2}\left( {1 + \frac{a}{\gamma_{C,1}}} \right)}A_{1}}} = 0$

that may also be written:

${\frac{^{2}A_{1}}{t^{2}} + {\Omega_{0,1}^{2} \cdot A_{1}}} = 0$

We can put a=a₀+α_(V) cos(2πf_(R)t) a₀ representing the low-frequencyuseful part of the acceleration applied to the sensor and α_(V) being aconstant representing the amplitude of the vibrations.

We put I=A₁ and we denote the Hilbert transform of I by Q. The spectrumof I is that of a signal modulated in frequency by a sine. We thereforehave a principal line centred on the frequency

$f_{a,0,1} = {\frac{\omega_{0,1}}{2\; \pi}\sqrt{1 + \frac{a_{0}}{\gamma_{C,1}}}}$

and secondary lines at f_(a0,1)±kf_(R).

By applying the Hilbert transform to relation 2bis, we then obtain:

${\frac{^{2}Q}{t^{2}} + {{\omega_{0,1}^{2}\left( {1 + \frac{a}{\gamma_{C,1}}} \right)}Q}} = 0$

By putting I=ρ cos(φ) and Q=ρ sin(φ), with ρ the instantaneous amplitudeof I and φ the instantaneous phase of I, in the above two differentialequations, we obtain the following two equations

${{\rho^{(2)}{\cos (\phi)}} - {2\rho^{(1)}\omega \; {\sin (\phi)}} - {\rho \left\lbrack {{\omega^{(1)}{\sin (\phi)}} + {\omega^{2}{\cos (\phi)}}} \right\rbrack}} = {{- {\omega_{0,1}^{2}\left( {1 + \frac{a}{\gamma_{C,1}}} \right)}}\rho \; {\cos (\phi)}}$${{\rho^{(2)}{\sin (\phi)}} + {2\rho^{(1)}\omega \; {\cos (\phi)}} + {\rho \left\lbrack {{\omega^{(1)}{\cos (\phi)}} - {\omega^{2}{\sin (\phi)}}} \right\rbrack}} = {{- {\omega_{0,1}^{2}\left( {1 + \frac{a}{\gamma_{C,1}}} \right)}}\rho \; {\sin (\phi)}}$

In this formulation and in those which follow, the simplified notationfor the derivatives is used, which consists in indicating the order ofdifferentiation by an exponent between brackets. By multiplying thefirst relation by cos(φ) and the second by sin(φ), by performing a sumand dividing by ρ, the instantaneous amplitude not vanishing, we obtain:

${\omega_{0,1}^{2}\left( {1 + \frac{a}{\gamma_{C,1}}} \right)} = {\omega^{2} - \frac{\rho^{(2)}}{\rho}}$

In the case of a vibrating barrel accelerometer, the output voltage U₁of the charge amplifier is proportional to the signal I. The spectrum ofthe oscillator position signal does not then comprise any low-frequencyterm. Denoting the instantaneous angular frequency of U₁ by ω₁ and itsinstantaneous amplitude by ρ₁, we have

$\omega_{1} = {{\omega \mspace{14mu} {and}\mspace{14mu} \frac{\rho_{1}^{(2)}}{\rho_{1}}} = {\frac{\rho^{(2)}}{\rho}.}}$

We thus obtain

${{Relation}\mspace{14mu} 6\mspace{14mu} {\omega_{0,1}^{2}\left( {1 + \frac{a}{\gamma_{C,1}}} \right)}} = {\omega_{1}^{2} - \frac{\rho_{1}^{(2)}}{\rho_{1}}}$

On the basis of the function

${\omega_{1}^{2} - \frac{\rho_{1}^{(2)}}{\rho_{1}}},$

of the values ω_(0,1) and γ_(c,1) situated in the left-hand side, it isthen easy to determine a which is the sought-after parameter.

Denoting the Hilbert transform of U₁ by V₁ and using the third propertyof the Hilbert transform, we then have:

${{Relation}\mspace{14mu} 7\mspace{14mu} {\omega_{0,1}^{2}\left( {1 + \frac{a}{\gamma_{C,1}}} \right)}} = {{\omega_{1}^{2} - \frac{\rho_{1}^{(2)}}{\rho_{1}}} = {- \frac{{U_{1}U_{1}^{(2)}} + {V_{1}V_{1}^{(2)}}}{U_{1}^{2} + V_{1}^{2}}}}$

The function

$\omega_{1}^{2} - \frac{\rho_{1}^{(2)}}{\rho_{1}}$

and hence the term a can thus be calculated by virtue of the function

$- {\frac{{U_{1}U_{1}^{(2)}} + {V_{1}V_{1}^{(2)}}}{U_{1}^{2} + V_{1}^{2}}.}$

The practical realization of this function

$- \frac{{U_{1}U_{1}^{(2)}} + {V_{1}V_{1}^{(2)}}}{U_{1}^{2} + V_{1}^{2}}$

poses no particular technical problems and is represented, by way ofexample, in the overview of FIG. 2.

In this figure, each rectangle represents an electronic elementaffording a particular function. Thus, the Hilbert transform V₁ isobtained by virtue of a finite impulse response filter denoted FIR_(TH)with N+1 points whose spectral band includes the lines of significantamplitude of the signal U₁. The derivatives of the signals arecalculated by a finite impulse response filter denoted FIR_(D) with L+1points whose spectral band also includes these lines. In order to ensurethe synchronization of the signals, the signal U₁ must be delayed byN/2points so as to be synchronized with the signal V₁ by delay linesdenoted R(N/2). Thereafter the signals U₁ and V₁ must be delayed byother delay lines denoted R(L) of L points so as to be synchronous withU₁ ⁽²⁾ and V₁ ⁽²⁾.

A wideband demodulation of the signal is thus achieved. The relation

${- \frac{{II}^{(2)} + {QQ}^{(2)}}{I^{2} + Q^{2}}} = {\omega^{2} - \frac{\rho^{(2)}}{\rho}}$

is always true regardless of I. The use of the algorithm employing thesecond derivative and the Hilbert transform with filters having a bigenough passband makes it possible to perform a wideband demodulation ofthe signal. The use of this algorithm is applicable to any dynamicsystem whose behaviour is governed by a law of the type of relation 0.The application of this algorithm to the VBA accelerometer with nononlinearity of order 3, makes it possible to obtain an item ofinformation regarding the instantaneous acceleration together with theuseful component and the vibrations. The useful component may beextracted therefrom by low-pass filtering.

It should be noted that by replacing I by ρ cos(φ) and Q by ρ sin(φ) intheir respective differential equations, we obtain 2ρω+ρω=0 i.e.ωρ²=constant. Any instantaneous angular frequency variation caused by avariation in the acceleration applied to the sensor gives rise to aninstantaneous amplitude variation. It is thus demonstrated that thesensor signal is not only modulated in frequency but also in amplitude.This amplitude modulation is not taken into account in the prior artprocedures, thus giving rise to an additional error.

The complete differential equation satisfied by the resonator is nowconsidered:

${\frac{^{2}A_{1}}{t^{2}} + {\frac{\omega_{0,1}}{Q_{R,1}}\frac{A_{1}}{t}} + {{\omega_{0,1}^{2}\left( {1 + \frac{a}{\gamma_{C,1}}} \right)}A_{1}} + {\beta_{i,1}A_{1}^{3}}} = {e_{1}(t)}$

The excitation term e₁ may be written e₁=K_(4,1)(V₀+v)²+K_(5,1)V₀ ² WithV₀: DC polarization voltage and

v: variable voltage obtained on the basis of phase- andamplitude-slaving to the charge amplifier output voltage representingthe sensor output signal. The voltage v therefore lies at the frequencyof the resonator.

It is assumed that the phase- and amplitude-slaving is done perfectly.The damping term may be eliminated. In fact we have:

${\frac{\omega_{0,1}}{Q_{R,1}}\frac{A_{1}}{t}} = {2\; K_{4,1}V_{0}v}$

The term v² possesses a component at twice the resonant frequency whichdoes not intervene significantly in the equation and a continuous termthat is negligible with respect to the square of the DC polarizationvoltage.

The equation thus becomes:

${{{Relation}\mspace{14mu} 8\mspace{11mu} \frac{^{2}A_{1}}{t^{2}}} + {{\omega_{0,1}^{2}\left( {1 + \frac{a}{\gamma_{C,1}}} \right)}A_{1}} + {\beta_{i,1}A_{1}^{3}}} = {K_{3,1}V_{0}^{2}}$with  K_(3, 1) = K_(5, 1) + K_(4, 1)

One seeks to decompose the solution of the motion into the form A₁=I+Δ

With I high-frequency signal whose spectrum is centred on

$f_{{a\; 0},1} = {\frac{\omega_{0,1}}{2\; \pi}\sqrt{1 + \frac{a_{0}}{\gamma_{C,1}}}}$

with lines at f_(a0,1)±kf_(R) and

Δ low-frequency term.

We reason firstly in the absence of nonlinearity of order 3. We then put

$\Delta = {\frac{K_{3,1}V_{0}^{2}}{\omega_{0,1}^{2}\left( {1 + \frac{a}{\gamma_{C,1}}} \right)}.}$

We consider the solution I of the equation

${\frac{^{2}I}{t^{2}} + {{\omega_{0,1}^{2}\left( {1 + \frac{a}{\gamma_{C,1}}} \right)}I}} = 0.$

I is a high-frequency signal whose spectrum is centred on f_(a0,1).Δ is a low-frequency frequency. It comprises lines at f_(R), 2f_(R) . .. due to the presence of vibrations in a, but these lines are ofnegligible amplitude. We therefore have

$\frac{^{2}\left( {I + \Delta} \right)}{t^{2}} \approx \frac{^{2}I}{t^{2}}$

In the absence of nonlinearity of order 3, we therefore decompose thesolution of equation (8) into a high-frequency term and a low-frequencyterm Δ defined by

$\Delta = \frac{K_{3,1}V_{0}^{2}}{\omega_{0,1}^{2}\left( {1 + \frac{a}{\gamma_{C,1}}} \right)}$

We neglect the effect of the nonlinearity on this low-frequency term.Indeed, according to the orders of magnitude involved, we have

${\omega_{0,1}^{2}\left( {1 + \frac{a}{\gamma_{C,1}}} \right)}\operatorname{>>}{{\beta_{i,1}\left( {I + \Delta} \right)}^{2}.}$

We therefore write A₁=I+Δ with I a high-frequency term and Δ alow-frequency term defined by

$\Delta = {\frac{K_{3,1}V_{0}^{2}}{\omega_{0,1}^{2}\left( {1 + \frac{a}{\gamma_{C,1}}} \right)}.}$

Equation 8 may be written:

${{{Relation}\mspace{14mu} 8\; {bis}\mspace{14mu} \frac{^{2}\left( {I + \Delta} \right)}{t^{2}}} + {{\omega_{0,1}^{2}\left( {1 + \frac{a}{\gamma_{C,1}}} \right)}\left( {I + \Delta} \right)} + {\beta_{i,1}\left( {I + \Delta} \right)}^{3}} = {K_{3,1}V_{0}^{2}}$

By applying the Hilbert transform to relation 8 bis with the samenotation as previously, we obtain, denoting the Hilbert transform of aΔby H(aΔ), relation 9 below:

$\begin{matrix}{{\frac{^{2}Q}{t^{2}} + {{\omega_{0,1}^{2}\left( {1 + \frac{a}{\gamma_{C,1}}} \right)}Q} + {\omega_{0,1}^{2}\frac{H\left( {a\; \Delta} \right)}{\gamma_{C,1}}} + {\beta_{i,1}\left( {\frac{Q^{3}}{2} + {\frac{3}{2}I^{2}Q} + {3\; \Delta \; {IQ}} + {3\; \Delta^{2}Q}} \right)}} = 0} & {{Relation}\mspace{14mu} 9}\end{matrix}$

By multiplying relation 8 bis by I and relation 9 by Q, we obtain:

${\omega_{0,1}^{2}\left( {1 + \frac{a}{\gamma_{C,1}}} \right)} = {{- \frac{{Q^{(2)}Q} + {I^{(2)}I}}{I^{2} + Q^{2}}} - \frac{\omega_{0,1}^{2}{H\left( {a\; \Delta} \right)}Q}{\gamma_{C,1}\left( {I^{2} + Q^{2}} \right)} - {\beta_{i,1}\left\lbrack {\frac{Q^{2}}{2} + I^{2} + {3\; \Delta \; I} + {3\; \Delta^{2}} + {\Delta^{3}\frac{I}{I^{2} + Q^{2}}}} \right\rbrack} + \frac{K_{3,1}V_{0}^{2}I}{I^{2} + Q^{2}} - {{\omega_{0,1}^{2}\left( {1 + \frac{a}{\gamma_{C,1}}} \right)}\frac{\Delta \; I}{I^{2} + Q^{2}}}}$

In the right-hand side of the equation, the terms H(aΔ) and aΔ areunknown. They are not taken into account in the algorithm insofar asthey cause a negligible error.

The terms in ΔI, Δ³I and K_(3,1)V₀ ²I are high-frequency terms. If theywere taken into account in the algorithm, they would in any event beeliminated by the low-pass filter placed at the end of the processing soas to isolate the useful part of the acceleration lying in the lowfrequencies. We have the following relation

${I^{2} + \frac{Q^{2}}{2}} = {{\frac{3}{4}\rho^{2}} + {\frac{\rho^{2}}{4}{{\cos \left( {2\; \phi} \right)}.}}}$

The term ρ² cos(2φ) does not lie in the useful frequency band either.

In view of these simplifications, the following relation is thenobtained:

${\omega_{0,1}^{2}\left( {1 + \frac{\hat{a}}{\gamma_{C,1}}} \right)} = {{- \frac{{Q^{(2)}Q} + {I^{(2)}I}}{I^{2} + Q^{2}}} - {\beta_{i,1}\left\lbrack {{\frac{3}{4}\rho^{2}} + {3\; \Delta^{2}}} \right\rbrack}}$

â is the acceleration estimated by the algorithm. It comprises thelow-frequency useful acceleration and the vibration terms.

Δ is estimated through the relation

${{- \frac{{I^{(2)}I} + {Q^{(2)}Q}}{I^{2} + Q^{2}}}\Delta} = {K_{3,1}V_{0}^{2}}$

The output voltage U₁ from the charge amplifier is proportional to thesignal I. U₁=K₁I. The information Δ having a low-frequency spectrum isnot present at the output of the charge amplifier. Relation 10 is thusobtained

$\Omega_{1}^{2} = {{- \frac{{V_{1}^{(2)}V_{1}} + {U_{1}^{(2)}U_{1}}}{U_{1}^{2} + V_{1}^{2}}} - {\beta_{i,1}\begin{bmatrix}{{\frac{3}{4\; K_{1}^{2}}\left( {U_{1}^{2} + V_{1}^{2}} \right)} +} \\{3\left( \frac{K_{3,1}V_{0}^{2}}{- \frac{{U_{1}U_{1}^{(2)}} + {V_{1}V_{1}^{(2)}}}{U_{1}^{2} + V_{1}^{2}}} \right)^{2}}\end{bmatrix}}}$

The relation may be rewritten by using the third property of the Hilberttransform, denoting the instantaneous angular frequency of U₁ by ω₁ andthe instantaneous amplitude of U₁ by ρ₁. Relation 11 is then obtained

$\Omega_{1}^{2} = {\omega_{1}^{2} - \frac{\rho_{1}^{(2)}}{\rho_{1}} - {\beta_{i,1}\left\lbrack {{\frac{3}{4\; K_{1}^{2}}\rho_{1}^{2}} + {3\left( \frac{K_{3,1}V_{0}^{2}}{\omega_{1}^{2} - \frac{\rho_{1}^{(2)}}{\rho_{1}}} \right)^{2}}} \right\rbrack}}$

The estimated acceleration is obtained from this information throughrelation 12

$\begin{matrix}{{\omega_{0,1}^{2}\left( {1 + \frac{\hat{a}}{\gamma_{C,1}}} \right)} = \Omega_{1}^{2}} & {{Relation}\mspace{14mu} 12}\end{matrix}$

The acceleration can thus be estimated by virtue of the function

$\omega_{1}^{2} - \frac{\rho_{1}^{(2)}}{\rho_{1}} - {\beta_{i,1}\left\lbrack {{\frac{3}{4\; K_{1}^{2}}\rho_{1}^{2}} + {3\left( \frac{K_{3,1}V_{0}^{2}}{\omega_{1}^{2} - \frac{\rho_{1}^{(2)}}{\rho_{1}}} \right)^{2}}} \right\rbrack}$

This function may be calculated by realizing the function

${- \frac{{V_{1}^{(2)}V_{1}} + {U_{1}^{(2)}U_{1}}}{U_{1}^{2} + V_{1}^{2}}} - {\beta_{i,1}\begin{bmatrix}{{\frac{3}{4\; K_{1}^{2}}\left( {U_{1}^{2} + V_{1}^{2}} \right)} +} \\{3\left( \frac{K_{3,1}V_{0}^{2}}{- \frac{{U_{1}U_{1}^{(2)}} + {V_{1}V_{1}^{(2)}}}{U_{1}^{2} + V_{1}^{2}}} \right)^{2}}\end{bmatrix}}$

The acceleration estimated by this algorithm does not contain anysignificant bias introduced by the vibrations. The information returnedby the algorithm varies linearly as a function of the accelerationapplied to the sensor.

Relative to the previous algorithm, two additional terms are seen toappear:

${- \frac{3}{4\; K_{1}^{2}}}{\beta_{i,1}\left( {U_{1}^{2} + V_{1}^{2}} \right)}$

a corrective term for the nonlinearity of order 3.

${- \beta_{i,1}}3\left( \frac{K_{3,1}V_{0}^{2}}{- \frac{{U_{1}U_{1}^{(2)}} + {V_{1}V_{1}^{(2)}}}{U_{1}^{2} + V_{1}^{2}}} \right)^{2}$

a corrective term for the DC polarization voltage.

Generally, two resonators having sensitive axes of like direction and ofopposite sense are employed on two different pathways. In this case,there are similar relations linking Ω to the other parameters on each ofthe pathways. On path 1, we have the previous relation 12 and on pathway2 relation 13 below, denoting the instantaneous angular frequency of U₂by ρ₂ and the instantaneous amplitude of U₂ by ω₂, U₂ being the voltageoutput by the filter on the second pathway

$\Omega_{2}^{2} = {\omega_{2}^{2} - \frac{\rho_{2}^{(2)}}{\rho_{2}} - {\beta_{i,2}\left\lbrack {{\frac{3}{4\; K_{2}^{2}}\rho_{2}^{2}} + {3\left( \frac{K_{3,2}V_{0}^{2}}{\omega_{2}^{2} - \frac{\rho_{2}^{(2)}}{\rho_{2}}} \right)^{2}}} \right\rbrack}}$

Using the third property of the Hilbert transform, it may be shown thatthe right-hand term of the above relation may be realized using U₂, itsHilbert transform denoted V₂ and their derivatives. The followingrelation is then obtained:

$\Omega_{2}^{2} = {{- \frac{{V_{2}^{(2)}V_{2}} + {U_{2}^{(2)}U_{2}}}{U_{2}^{2} + V_{2}^{2}}} - {\beta_{i,2}\begin{bmatrix}{{\frac{3}{4\; K_{2}^{2}}\left( {U_{2}^{2} + V_{2}^{2}} \right)} +} \\{3\left( \frac{K_{3,2}V_{0}^{2}}{- \frac{{U_{2}U_{2}^{(2)}} + {V_{2}V_{2}^{(2)}}}{U_{2}^{2} + V_{2}^{2}}} \right)^{2}}\end{bmatrix}}}$

It may be shown in the same way that the acceleration may be estimatedon this pathway by virtue of the relation:

$\begin{matrix}{{\omega_{0,2}^{2}\left( {1 - \frac{\hat{a}}{\gamma_{C,2}}} \right)} = \Omega_{2}^{2}} & {{Relation}\mspace{14mu} 15}\end{matrix}$

Starting from these two relations, it is possible to perform adifferential processing, that is to say a processing which uses theinformation of the two pathways of the sensor and which determines onthe basis of the information returned by each pathway not only theacceleration information but in addition the drifting of the mainparameters due in particular to temperature.

The simple difference of the two items of information may be used, itbeing possible for the acceleration to be estimated by virtue of therelation

${\left( {\frac{\omega_{0,1}^{2}}{\gamma_{C,1}} + \frac{\omega_{0,2}^{2}}{\gamma_{C,2}}} \right)\hat{a}} = {\Omega_{1}^{2} - \Omega_{2}^{2} - \left( {\omega_{0,1}^{2} - \omega_{0,2}^{2}} \right)}$

However, if the terms in ω₀ and in γ_(C) drift and are dependent on aparameter such as the temperature, then poor estimation of saidtemperature then causes:

-   -   a bias error through poor estimation of the term ω_(0,1)        ²-ω_(0,2) ²    -   a scale factor error through poor estimation of the term

$\left( {\frac{\omega_{0,1}^{2}}{\gamma_{C,1}} + \frac{\omega_{0,2}^{2}}{\gamma_{C,2}}} \right).$

These difficulties may in particular be circumvented by using a moresuitable differential processing. It was seen that the acceleration wasconnected to the information returned on the two pathways by relations11 and 13. The following two quantities are estimated:

-   -   weighted difference of the two items of information

$\Omega_{1}^{2} - {\frac{\omega_{0,1}^{2}}{\omega_{0,2}^{2}}\Omega_{2}^{2}}$

-   -   weighted sum of the two items of information

$\Omega_{1}^{2} + {\frac{\omega_{0,1}^{2}}{\omega_{0,2}^{2}}\frac{\gamma_{C,2}}{\gamma_{C,1}}\Omega_{2}^{2}}$

We then have the following relations:

$\begin{matrix}{{\Omega_{1}^{2} - {\frac{\omega_{0,1}^{2}}{\omega_{0,2}^{2}}\Omega_{2}^{2}}} = {{\omega_{0,1}^{2}\left( {\frac{1}{\gamma_{c,1}} + \frac{1}{\gamma_{C,2}}} \right)}\hat{a}}} & {{Relation}\mspace{14mu} 16} \\{{\Omega_{1}^{2} + {\frac{\omega_{0,1}^{2}}{\omega_{0,2}^{2}}\frac{\gamma_{C,2}}{\gamma_{C,1}}\Omega_{2}^{2}}} = {\omega_{0,1}^{2}\left( {1 + \frac{\gamma_{C,2}}{\gamma_{C,1}}} \right)}} & {{Relation}\mspace{14mu} 17}\end{matrix}$

The estimated acceleration of relation 16 then comprises the useful partand the vibration terms.

By assuming that the critical acceleration and that the rest frequencyvary according to an exponential law as a function of temperature, theratios

$\frac{\omega_{0,1}^{2}}{\omega_{0,2}^{2}}\frac{\gamma_{C,2}}{\gamma_{C,1}}\mspace{14mu} {and}{\mspace{11mu} \;}\frac{\omega_{0,1}^{2}}{\omega_{0,2}^{2}}$

are independent of temperature and may also be calibrated. The weightedsum of the two items of information therefore theoretically depends onlyon the temperature and does not depend on the acceleration. By virtue ofthe weighted sum of Ω₁ ² and of Ω² ₂ of relation 17, it is thus possibleto estimate the temperature and then determine the acceleration byvirtue of relation 16. The weighted difference thus makes it possible tocircumvent the bias error due to poor estimation of the temperature.

These two relations may be rewritten by expressing the physicalparameters as temperature dependent polynomials. It is assumed that theweighted sum depends on the temperature and also on the acceleration. Wethus obtain

$\begin{matrix}{{\left( {\alpha_{0} + {\alpha_{1}T} + {\alpha_{2}T^{2}}} \right)\hat{a}} = {\Omega_{1}^{2} - {\frac{\omega_{0,1}^{2}}{\omega_{0,2}^{2}}\Omega_{2}^{2}}}} \\{{\delta_{0} + {\delta_{1}T} + {\delta_{2}T^{2}} + {\delta_{3}\hat{a}} + {\delta_{4}\hat{a}T}} = {\Omega_{1}^{2} + {\frac{\omega_{0,1}^{2}}{\omega_{0,2}^{2}}\frac{\gamma_{{C,2}\;}}{\gamma_{C,1}}\Omega_{2}^{2}}}}\end{matrix}$

The coefficients α_(i) and δ_(i) are known by calibration. The usefulacceleration and the temperature may therefore be deduced from theinformation Ω₁ ² and Ω₂ ².

Practical realization does not pose any particular problems. By way ofexample, the differential processing is represented in FIG. 4. Theweighted sum and the difference are filtered by a band-stop filterdenoted B.S.F. centred on the frequency of the vibrations and by alow-pass filter denoted L.P.F. The acceleration estimated by virtue ofthese relations at the output of the filters is then solely the usefulacceleration.

It will be readily seen by one of ordinary skill in the art that thepresent invention fulfils all of the objects set forth above. Afterreading the foregoing specification, one of ordinary skill in the artwill be able to affect various changes, substitutions of equivalents andvarious aspects of the invention as broadly disclosed herein. It istherefore intended that the protection granted hereon be limited only bydefinition contained in the appended claims and equivalents thereof.

1. System for analysing the oscillation frequency of a device vibratingalong an axis, said system comprising means for measuring the position Aof the device along this axis, the signal emanating from said means andrepresentative of the position A being represented by a time dependentfunction U, said system comprising first means making it possible tocalculate the function${\omega_{1}^{2} - \frac{\rho_{1}^{(2)}}{\rho_{1}}},\omega_{1}$ beingthe instantaneous angular frequency of U, ρ₁ its instantaneous amplitudeand ρ₁ ⁽²⁾ the second derivative of said instantaneous amplitude, thefunction $\omega_{1}^{2} - \frac{\rho_{1}^{(2)}}{\rho_{1}}$ beingrepresentative of the square of the resonant angular frequency Ω of thevibrating device.
 2. Analysis system according to claim 1, wherein thesaid first means comprise means for calculating the second derivative ofthe function U denoted U⁽²⁾, the Hilbert transform of the function Udenoted V, the second derivative of this Hilbert transform denoted V⁽²⁾as well as a first function equal to$- \frac{{U.U^{(2)}} + {V.V^{(2)}}}{U^{2} + V^{2}}$ mathematically equalto ${\omega_{1}^{2} - \frac{\rho_{1}^{(2)}}{\rho_{1}}},$ representativeof the square of the resonant angular frequency Ω of the vibratingdevice.
 3. Analysis system according to claim 2, wherein, when thedifferential equation representative of the variations in the position Aof the device comprises nonlinearity terms of order 3, said device thencomprises second means making it possible to realize, using the previousnotation, a second function equal to${{- \frac{{V^{(2)}V} + {U^{(2)}U}}{U^{2} + V^{2}}} - {\beta_{i}\left\lbrack {\frac{3}{4K_{1}^{2}}\left( {U^{2} + V^{2}} \right)} \right\rbrack}},$representative of the square of the resonant angular frequency Ω of thevibrating device, β_(i) and K₁ being constants.
 4. Analysis systemaccording to claim 3, wherein, when the vibrating device has forcedoscillations, that is to say the device comprises a phase- andamplitude-slaving disposed in such a way as to eliminate the naturaldamping of the device, said slaving generating a polarization V₀, saiddevice then comprises third means making it possible to realize, usingthe previous notation, a third function equal to${{- \frac{{V^{(2)}V} + {U^{(2)}U}}{U^{2} + V^{2}}} - {\beta_{i}\left\lbrack {{\frac{3}{4K_{1}^{2}}\left( {U^{2} + V^{2}} \right)} + {3\left( \frac{K_{3}V_{0}^{2}}{- \frac{{UU}^{(2)} + {VV}^{(2)}}{U^{2} + V^{2}}} \right)^{2}}} \right\rbrack}},$representative of the square of the resonant angular frequency Ω of thevibrating device, β_(i), K₁ and K₃ being constants.
 5. Analysis systemaccording to claim 1, wherein the square of the resonant angularfrequency is linked to a parameter a to be measured by the relation:$\Omega^{2} = {{\omega_{0}^{2}\left( {1 + \frac{a}{\gamma_{C}}} \right)}\gamma_{C}}$being a constant, ω₀ being the initial angular frequency in the absenceof said parameter.
 6. Analysis system according to claim 5, wherein thevibrating device comprising two identical means of vibration, each ofthe means being connected to the analysis system, said systemcomprising: measurement means able to calculate on the one hand, eithera first function or a second function or a third function representativeof the square of the resonant angular frequency Ω₁ of the first means ofvibration and on the other hand either a first function or a secondfunction or a third function representative of the square of theresonant angular frequency Ω₂ of the second means of vibration; meansfor calculating the following functions:$\Omega_{1}^{2} - {\frac{\omega_{01}^{2}}{\omega_{02}^{2}}\Omega_{2}^{2}\mspace{14mu} {and}\mspace{14mu} \Omega_{1}^{2}} + {\frac{\omega_{0,1}^{2}}{\omega_{0,2}^{2}}\frac{\gamma_{C,2}}{\gamma_{C,1}}\Omega_{2}^{2}}$using the same notation as previously, the indices representing thefirst or the second means of vibration.
 7. Analysis system according toclaim 2, wherein the analysis system is an electronic system, thefunction U being an electrical parameter, said system comprising meansof digitizing and sampling the function U, first finite impulse responsefilters able to realize the Hilbert transform of an electronic function,second finite impulse response filters able to realize the derivative ofan electrical function, delay lines making it possible to synchronizethe various sampled signals, electronic means realizing the functions ofsummations, multiplication and division and band-stop filters andlow-pass filters.
 8. Analysis system according to claim 3, wherein theanalysis system is an electronic system, the function U being anelectrical parameter, said system comprising means of digitizing andsampling the function U, first finite impulse response filters able torealize the Hilbert transform of an electronic function, second finiteimpulse response filters able to realize the derivative of an electricalfunction, delay lines making it possible to synchronize the varioussampled signals, electronic means realizing the functions of summations,multiplication and division and band-stop filters and low-pass filters.9. Analysis system according to claim 4, wherein the analysis system isan electronic system, the function U being an electrical parameter, saidsystem comprising means of digitizing and sampling the function U, firstfinite impulse response filters able to realize the Hilbert transform ofan electronic function, second finite impulse response filters able torealize the derivative of an electrical function, delay lines making itpossible to synchronize the various sampled signals, electronic meansrealizing the functions of summations, multiplication and division andband-stop filters and low-pass filters.
 10. Analysis system according toclaim 5, wherein the analysis system is an electronic system, thefunction U being an electrical parameter, said system comprising meansof digitizing and sampling the function U, first finite impulse responsefilters able to realize the Hilbert transform of an electronic function,second finite impulse response filters able to realize the derivative ofan electrical function, delay lines making it possible to synchronizethe various sampled signals, electronic means realizing the functions ofsummations, multiplication and division and band-stop filters andlow-pass filters.
 11. Analysis system according to claim 6, wherein theanalysis system is an electronic system, the function U being anelectrical parameter, said system comprising means of digitizing andsampling the function U, first finite impulse response filters able torealize the Hilbert transform of an electronic function, second finiteimpulse response filters able to realize the derivative of an electricalfunction, delay lines making it possible to synchronize the varioussampled signals, electronic means realizing the functions of summations,multiplication and division and band-stop filters and low-pass filters.